Participants
Dept. of Mathematical Sciences, Aalborg University: E.Susanne Christensen, Susanne G. Bøttcher
Dept. of Forensic Genetics, University of Copenhagen: Niels Morling, Helle Smidt Mogensen
Dept. of Statistics, Oxford University: Steffen L. Lauritzen
Abstract
This project investigates the behavior of the PCR
Amplification Kit. A number of known DNA-profiles are
mixed two by two in "known” proportions and
analyzed.
Short Tandem Repeats (STR)
Human identity tests focus on Short Tandem Repeat markers
(STR markers). STR markers are genetic loci consisting of
repeated subunits, 2-8 base pairs in length. Discrimination
between individuals is possible because the number of
subunits present for a given marker varies from person to
person. Simultaneous analysis of several STR markers allows
for the compilation of a profile, which is almost unique to a
given individual.
Gamma distribution models are fitted to the resulting
data to learn to what extent actual mixing proportions
can be rediscovered in the amplifier output and
thereby the question of confidence in separate DNA profiles suggested by an output is addressed.
Weight of DNA- British data
vWA
1
-1
0
1
0
-1
std residual
DNA mixture output
std residual
2
2
FGA
1.0
1.5
0.5
1.0
weight
weight
D3
D18
1.5
0.53 x 10ˉ3
3.38 x 10ˉ4
2.95 x 10ˉ4
2.53 x 10ˉ4
1.80 x 10ˉ5
2.90 x 10ˉ4
2.50 x 10ˉ4
7.13 x 10ˉ5
5.42 x 10ˉ4
1.70 x 10ˉ5
0.5
1.0
2
1
1.5
0.5
weight
Amount of DNA- Danish data
FGA
vWA
0
0
DNA amount
DNA amount
D3
D18
Er
Ly
Va
Ca
Er
Ly
Va
Er
Ly
Va
Std. residuals
4
2
Bi
Er
Ly
Va
-2
-2
0
Std. residuals
1.0
0.5
0.0
Bi
4
6
Mix with Ca, vWA
50
30
10
-0.4
-10
0
-10
Ca
100 200 300 400 500
2
Mix with Ca, FGA
100 200 300 400 500
0
Mix with Bi, vWA
0
100 200 300 400 500
0
100
DNA amount
Mix with Bi, D18
500
DNA amount
6
Discussion
Er
Ly
Va
4
2
0
0
-2
-1
-2
Ca
Ca
Er
Ly
Va
Bi
Mix with Er, vWA
Er
Ly
Va
Va
Ca
Ly
Va
1.0
Ca
Er
Va
Ca
Er
Va
4
Mix with Ly, D18
3
4
4
Bi
Mix with Ly, D3
6
1
0
-1 0
2
2
2
2
4
Mix with Ly, vWA
0.0
Bi
Mix with Er, D18
6
Mix with Er, D3
Va
0
Bi
6
Ly
Ly
0.5
40
20
0.5
0.0
Ca
Er
60
1.0
60
40
20
0
Bi
Bi
Mix with Ly, FGA
80
80
Mix with Er, FGA
-2
0
0
Data were created by an controlled laboratory
experiment performed by Section of Forensic Genetics,
University of Copenhagen to investigate the
performance of the AmpF1STRSGM Plus PCR
Amplification Kit (Applied Biosystems, CA, USA) in STRprofiling. Samples were created from 4 persons with
known profiles. Mixtures of two contributors with
different but known amounts of DNA were created as a
full factor experiment. Also one-contributor samples
were analyzed for all four persons in different
concentrations. Every sample were analyzed twice.
Observations used from this dataset are peak height,
which are approximately proportional to peak area in
this data.
-2
The Cph-Crime-SGMP-Mix-Exp-2005-1 dataset
-1 0 1
2
2
0
200 300 400
Mix with Ca, D18
6
Mix with Ca, D3
3 4
Mix with Bi, D3
1.5
weight
6
Mix with Bi, FGA
1.0
1.0
1.43 x 10ˉ3
1.20 x 10ˉ3
1.09 x 10ˉ3
7.73 x 10ˉ4
1.80 x 10ˉ5
1.29 x 10ˉ3
1.13 x 10ˉ3
4.69 x 10ˉ5
1.24 x 10ˉ3
2.40 x 10ˉ5
0.5
x 10ˉ2
x 10ˉ3
x 10ˉ3
x 10ˉ3
x 10ˉ4
x 10ˉ3
x 10ˉ3
x 10ˉ3
x 10ˉ2
x 10ˉ4
0.0
1.04
6.90
5.93
4.92
1.90
6.39
5.70
5.48
1.20
2.06
Std. residuals
10ˉ2
10ˉ2
10ˉ2
10ˉ2
10ˉ3
10ˉ2
10ˉ1
10ˉ2
10ˉ2
10ˉ4
80
x
x
x
x
x
x
x
x
x
x
60
2.88
2.53
2.26
1.57
0.53
2.46
0.26
2.29
2.84
6.93
40
7.99 x 10ˉ6
6.33 x 10ˉ6
2.38 x 10ˉ5
1.44 x 10ˉ5
1.18 x 10ˉ5
8.35 x 10ˉ6
5.03 x 10ˉ6
6.99 x 10ˉ6
1.24 x 10ˉ5
5.16 x 10ˉ6
Std. residuals- Danish data
1
The constructed two-persons mixtures divides the
persons in two groups, forming the mixtures: G.J and
C.J, (involving three persons), and C.A, S.A, H.A and
C.P, (involving four other persons). Observations in this
British data are peak area.
x 10ˉ1
x 10ˉ1
x 10ˉ1
x 10ˉ1
x 10ˉ1
x 10ˉ1
x 10ˉ1
x 10ˉ1
x 10ˉ1
x 10ˉ1
std .err ( ˆ )
20
1.60
2.35
4.67
2.26
3.29
2.20
1.82
1.93
2.03
2.28
std.err (ˆ )
0
10ˉ5
10ˉ5
10ˉ6
10ˉ5
10ˉ4
10ˉ5
10ˉ5
10ˉ5
10ˉ5
10ˉ5
ˆ
̂
Std. residuals
x
x
x
x
x
x
x
x
x
x
10
Data have been processed by Genescan™ software,
(Applied Biosystems). Systems analyzed are: D3S1358,
vWA, D16S539, D2S1338, AMEL, D8S1179, D21S11,
D18S51, D19S433,THO1,FGA.
5.55
4.30
2.05
5.63
1.02
6.00
4.30
5.05
9.01
3.29
system
THO
D16
D18
D19
D2
D21
D3
D8
FGA
vWA
4
Mixtures have been created from two contributors with
known DNA profile, known mixture rates and known
concentrations at 0.5 ng/µl, and seven different mixing
proportions.
1.44
1.90
4.50
1.88
3.26
1.91
2.06
1.75
1.83
1.83
std .err ( ˆ )
std.err (ˆ )
ˆ
0.0 0.2 0.4
The QLB dataset
̂
20
Source: Forensic DNA typing. J. M. Butler
system
THO
D16
D18
D19
D2
D21
D3
D8
FGA
vWA
-2
-1
-1
0
1
std residual
Estimates- Danish data
Estimates- British data
0
std residual
2
3
3
4
0.5
Bi
Ca
Ly
Va
Bi
Ca
Ly
Bi
Va
Ca
Er
Va
Bi
Ca
Er
Va
Std. residuals- British data
Notation
m: system, e.g. FGA
i : person
j : sample
a : type of allel
nm( j): number of allel type a (0, 1 or 2)
ia
wm( j): peak height or peak area
ia
 ( j): "contribution" of DNA
i
m( j)  wm ( j)
 wia
i
a
kam( j)   ( j)nm( j)
ia
i i
Residuals reveals the need for incorporation of a personal
effect in the model, beyond what is included in amount of
DNA present. The additional effect is seen in both onecontributor samples and in mixtures.
There is no obvious difference in using mixing proportions
or actual amount of DNA in the parameterization. In both
cases it is seen, that some moderation from proportionality
should be considered as residuals decrease with increasing
amount of DNA in the sample.
References
L( W | , n , ,  )  ( n ,  ),
ia i ia
i ia
  1 w

f (w) 
w e
( )
E(W | , na, ,  )   n 
ia
i ia
V(W | , na, ,  )  n  2  E(W | , na, ,  ) 
ia
i ia
ia
This ensures that:





i ( j)nia



a
i
j
ˆ
 ˆ
a  j wa ( j)





J. Mortera, A.P.David, S.L. Lauritzen, Probabilistic expert systems for
DNA mixture profiling, Theor. Popul. Biol. 63 (2003), 1919-205
M.W. Perlin, B. Szabady, Linear mixture analysis, a mathematical
approach to resolving mixed DNA samples, J. Forensic Sci. 45 (2001),
1372-1378
The parameters are estimated by solving the following
maximum likelihood equations,





B.S. Weir, C.M. Triggs, L.Starling, L.I. Stowell, K.A.J. Walsh, J.S.
Buckleton, Interpreting DNA mixtures, J.Forensic Sci. 42(5), (1999),
987-995
R.G Cowell, S.L.Lauritzen, J.Mortera, Identification and separation of
DNA mixtures using peak area information, Forensic Sci. Int. To appear.
Estimation















 i ( j)nia  ˆ  (i ( j)nia )   i ( j)nia (ln wa  ln ˆ ),
a j i
a j i
i
where  is the digamma function.
From the Danish data it is clear that a full understanding
and modeling of the STR- amplification still is far from being
obtained. It is however of great importance that this work is
carried out in order to address the question of separation of
DNA profiles from mixture samples.
J.M Butler, Forensic DNA typing. Elsevier, USA. 2005
with mean and variance given as:





The findings in Cph-Crime-SGMP-Mix-Exp-2005-1 data are
not quite as consistent, but there is a clear tendency of
over- or underestimation throughout the systems for any
two persons, as there are systematic differences between
individuals analyzed. The tendency to decreasing std.
residuals for increase in amount in DNA is found in the
Danish data as well.
The model seems logical and produces a fair fit to data.
The model is defined as follows:
L(Wa | , na, ,  )     n , 
i i ia
In the QLB-data mixture C.J was consistently overestimated,
S.A was overestimated in 9 of 10 markers, H.A was
underestimated for 6 markers and mainly so for 3 others,
and the C.P mixture was consistently underestimated for 2
markers and mainly underestimated for 3 markers. There is
a slight tendency to decreasing std. residuals for increasing
weight of DNA in the sample.
Conclusion
Model





The plots shows results from four systems only.
T. Wang, N. Xue, R. Wickenheiser, Least square deconvolution (LSD); a
new way of resolving STR/DNA mixture samples, in: Proceedings of the
13’th International Symposium on Human Identification, October 7-10,
Phoenix, AZ, 2002.
I. Evett, P.Gill, J. Lambert, Taking account of peak ares when
interpreting mixed DNA profiles, J. Forensic Sci, 43 (1998), 62-69
Future works
A parameter of ’personal impact’ shall be incorporated in the
model.
The important question of ’drop-outs’, ’drop-ins’ and stutters
shall be considered.
Estimation of combined ’personal impact and amount DNA’
shall be considered for the purpose of separation of DNA –
profiles in mixture DNA samples from crime scenes on basis
of the model.
Sensibility to the actual machine should be investigated, and
influence of injection times be considered. A calibration
algorithm for adaption to machinery will be constructed.
Contact:
E. Susanne Christensen, e-mail: susanne@math.aau.dk